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                                                            J. Math. Anal. Appl. 316 (2006) 753–763
                                                                                                                        www.elsevier.com/locate/jmaa
                                    Aniterative method for solving nonlinear
                                                               functional equations
                                                  Varsha Daftardar-Gejji∗, Hossein Jafari
                                       Department of Mathematics, University of Pune, Ganeshkhind, Pune 411007, India
                                                                         Received 17 February 2005
                                                                        Available online 9 June 2005
                                                                          Submitted by M. Iannelli
                      Abstract
                         An iterative method for solving nonlinear functional equations, viz. nonlinear Volterra integral
                      equations, algebraic equations and systems of ordinary differential equation, nonlinear algebraic
                      equations and fractional differential equations has been discussed.
                      2005Elsevier Inc. All rights reserved.
                      Keywords: Nonlinear Volterra integral equations; System of ordinary differential equations; Iterative method;
                          Contraction; Banach fixed point theorem; Fractional differential equation
                      1. Introduction
                          Avariety of problems in physics, chemistry and biology have their mathematical set-
                      ting as integral equations [10]. Therefore, developing methods to solve integral equations
                      (especially nonlinear), is receiving increasing attention in recent years [1–8]. In the present
                      work we describe an iterative method which can be utilized to obtain solutions of nonlin-
                      ear functional equations. The method when combined with algebraic computing software
                      (Mathematica, e.g.) turns out to be powerful.
                       * Corresponding author.
                          E-mail addresses: vsgejji@math.unipune.ernet.in (V. Daftardar-Gejji), jafari_h@math.com (H. Jafari).
                      0022-247X/$ – see front matter  2005 Elsevier Inc. All rights reserved.
                      doi:10.1016/j.jmaa.2005.05.009
          754         V. Daftardar-Gejji, H. Jafari / J. Math. Anal. Appl. 316 (2006) 753–763
            The present paper has been organized as follows. In Section 2 the iterative method is
          described and in Section 3 existence of solutions for nonlinear Volterra integral equations
          has been proved using this method. Illustrative examples have been presented in Section 4
          followed by the conclusions in Section 5.
          2. Aniterative method
            Consider the following general functional equation:
               y =N(y)+f,                                                  (1)
          where N is a nonlinear operator from a Banach space B →B and f is a known function.
          Wearelookingforasolution y of Eq. (1) having the series form:
                   ∞
               y =
yi.                                                     (2)
                  i=0
          Thenonlinear operator N can be decomposed as
                 ∞             ∞  i         i−1  
               N 
y =N(y)+
 N 
y −N 
y .                                   (3)
                      i      0             j           j
                  i=0            i=1    j=0        j=0
          FromEqs.(2)and(3), Eq. (1) is equivalent to
               ∞                ∞  i         i−1   
               
y =f+N(y)+
 N 
y −N 
y                   .                 (4)
                  i         0             j           j
               i=0              i=1    j=0        j=0
          Wedefinetherecurrence relation:
               y =f,
               y0=N(y),                                                   (5)
                1      0
                 y   =N(y +···+y )−N(y +···+y       ),  m=1,2,....
                 m+1      0       m      0       m−1
          Then
               (y +···+y   ) =N(y +···+y ),   m=1,2,...,                   (6)
                1       m+1      0        m
          and
                      ∞
               y =f +
yi.                                                  (7)
                      i=1
          If N is a contraction, i.e. N(x)−N(y)
Kx −y, 00:
                    y′ =−y2,y(1)=1.                                                                  (11)
              Theinitial value problem in Eq. (11) is equivalent to the integral equation
                                 x
                    y(x)=1− y2dt.                                                                   (12)
                                1
              Following the algorithm given in (9):
                    y0 =1,
                                      x
                    y                   2
                       =N(y)=− y dt=1−x,
                     1        0          0
                                     1
                                                  4            2    x3
                    y =N(y +y )−N(y )= −3x+2x −                       ,
                     2        0    1        0     3                 3
                          113    64x    34x2     85x3     41x4    4x5     2x6    x7
                    y3 = 63 − 9 + 3 − 9 + 9 − 3 + 9 −63,
                      .
                      .
                      .
                                       

              In Fig. 1 we have plotted   5   y (x), which is almost equal to the exact solution y = x−1.
                                          i=0 i
                                                         Fig. 1.